All of the 5th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$3.00$ each for students, and the group paid $$37.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$30.00$ each for teachers and $$9.50$ each for students, and the group paid $$137.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3y = 37.5}$ ${30x+9.5y = 137.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-12y = -150}$ ${30x+9.5y = 137.5}$ Add the top and bottom equations together. $ -2.5y = -12.5 $ $ y = \dfrac{-12.5}{-2.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {7.5x+3y = 37.5}$ to find $x$ ${7.5x + 3}{(5)}{= 37.5}$ $7.5x+15 = 37.5$ $7.5x = 22.5$ $x = \dfrac{22.5}{7.5}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {30x+9.5y = 137.5}$ and get the same answer for $x$ ${30x + 9.5}{(5)}{= 137.5}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.